Noise correction of large deviations with anomalous scaling

We present a path integral calculation of the probability distribution associated with the time-integrated moments of the Ornstein-Uhlenbeck process that includes the Gaussian prefactor in addition to the dominant path or instanton term obtained in the low-noise limit. The instanton term was obtained recently [D. Nickelsen and H. Touchette, Phys. Rev. Lett. 121, 090602 (2018)] and shows that the large deviations of the time-integrated moments are anomalous in the sense that the logarithm of their distribution scales nonlinearly with the integration time. The Gaussian prefactor gives a correction to the low-noise approximation and leads us to define an instanton variance giving some insights as to how anomalous large deviations are created in time. The results are compared with simulations based on importance sampling, extending our previous results based on direct Monte Carlo simulations. We conclude by explaining why many of the standard analytical and numerical methods of large deviation theory fail in the case of anomalous large deviations.

Automated summary

In this article, we present a path integral calculation of the probability distribution associated with the time-integrated moments of the Ornstein-Uhlenbeck process. We discovered that the logarithm of the distribution of these moments scales nonlinearly with the integration time, indicating anomalous large deviations. By introducing a Gaussian prefactor and defining an instanton variance, we gain insights into how these anomalous large deviations are generated in time. Our results are compared with simulations based on importance sampling, and we explain why standard analytical and numerical methods fail in the case of anomalous large deviations.